Structure of collective states built on the $11/2^{+}$ isomer in ${}^{187}\mathrm{Os}$: Quasiparticle-plus-triaxial-rotor model and interpretation as tilted-precession bands

Background: The shape of most nuclei is described by its quadrupole deformation (showing the deviation from spherical shape) and its triaxiality (showing the deviation from axial symmetry). Nuclei affected by triaxiality show additional collective rotational bands, called $\gamma$ bands, that appear at low excitation energy. The $\gamma$ bands can be caused by the precession of a nucleus with triaxial shape, but can also arise from small $\gamma$ vibrations around an axially symmetric shape.

Purpose: The aim of this work is to search for new collective excitations in ${}^{187}\mathrm{Os}$ in particular related with the $\gamma$ degree of freedom.

Methods: The rotational structures of ${}^{187}\mathrm{Os}$ were populated by the ${}^{186}\mathrm{W}({}^4\mathrm{He},3\mathrm{n}){}^{187}\mathrm{Os}$ reaction at a beam energy of 37 MeV. The measurements of $\gamma$ - $\gamma$ coincidences, angular distribution ratios and $\gamma$-ray intensities were performed using eleven Compton-suppressed Ge clover detectors.

Results: The previously known positive-parity band built on the $11/2^{+}$ isomer has been extended up and a new excited positive-parity band built above a $15/2^{+}$ state has been observed. The $11/2^{+}$ band was assigned a $\nu i_{13/2}$ configuration while the new $15/2^{+}$ band was associated with a coupling of the valence $i_{13/2}$ neutron with the $\gamma$ band of the even-even core. The quasiparticle-plus-triaxial-rotor model calculations provide a good agreement with the experimental data for both bands. They describe the $15/2^{+}$ band as a collective excitation with respect to the $11/2^{+}$ band that corresponds to a precession of the intermediate nuclear axis similarly to the precession of a rotating top.

Conclusions: As shown by the calculations, the new rotational band can be understood as resulting from the three-dimensional rotation of a triaxially-deformed nucleus. However, a description based on the vibrations of a $\gamma$-soft nuclear shape should also be investigated in order to firmly establish the nature of the excited positive-parity band. Further studies able to distinguish between these alternative descriptions will be beneficial.

Figure 1

Sketch illustrating the precession of the rotational angular momentum $R$ around the intermediate axis of a triaxially-deformed nucleus. For the ground-state band the total rotational angular momentum $R$ is approximately aligned along the intermediate axis, however it is tilted away from this axis and precesses around it for the $\gamma$ bands, because these bands involve unfavoured rotation in the plane defined by the short and long nuclear axes.

Figure 2

Summary of collective excitations in the even-even stable osmium isotopes showing the ground state and the excited $0_2^{+}, 2^{+}$, and $4^{+}$ rotational bands [10, 11, 12, 13, 14, 15].

Figure 3

Partial level scheme of ${}^{187}\mathrm{Os}$ deduced from the current work showing the positive-parity bands. New transitions are labeled in red and asterisk (*) symbol while previously known transitions are labeled in black.

Figure 4

Coincidence spectrum obtained by setting a gate on the 654.1-keV transition of band 1. New transitions are labeled in red and asterisk (*), while contaminants and other transitions of ${}^{187}\mathrm{Os}$ not associated with the band of interest are denoted by green and hash (#) symbol. The known transitions associated with the band of interest are labeled in black. The spectrum gated on 782.2 keV is inserted to show the 475.5- and 654.1-keV transitions.

Figure 5

Coincidence spectrum obtained by setting a gate on the 686.4-keV transition of band 2. New transitions are labeled in red and asterisk (*), while contaminants and other transitions of ${}^{187}\mathrm{Os}$ not associated with the band of interest are denoted by green and hash (#) symbol. The known transitions associated with the band of interest are labeled in black. The spectrum gated on 724.7 keV is inserted to show the 686.4-keV transition.

Figure 6

Coincidence spectrum obtained by setting a gate on the 573.5-keV transition of band 3. New transitions are labeled in red and asterisk (*), while contaminants and other transitions of ${}^{187}\mathrm{Os}$ not associated with the band of interest are denoted by green and hash (#) symbol. The spectrum gated on 487.1 keV is inserted to show the 573.5-keV transition.

Figure 7

Coincidence spectrum obtained by setting a gate on the 520.0-keV transition of band 4. New transitions are labeled in red and asterisk (*), while contaminants and other transitions of ${}^{187}\mathrm{Os}$ not associated with the band of interest are denoted by green and hash (#) symbol. The spectrum gated on 695.5 keV is inserted to show the 520.0-, 736.0-, 828.2-, and 954.1-keV transitions.

Figure 8

Compilation of the experimental excitation energies with respect to a rigid rotor reference for the ground state bands and the $\gamma$ bands of ${}^{182}\mathrm{Os}$ [10], ${}^{184}\mathrm{Os}$ [11], ${}^{186}\mathrm{Os}$ [12] and ${}^{188}\mathrm{Os}$ [13]. The even- and odd-spin sequences of the $\gamma$ bands are labeled with open and closed symbols, respectively, and the same colours.

Figure 9

Experimental excitation energies relative to a rigid rotor reference for the positive-parity bands in ${}^{187}\mathrm{Os}$ in comparison with the corresponding bands in ${}^{183}\mathrm{Os}$ [43] and ${}^{185}\mathrm{Os}$ [44]. The signature partner bands are labeled in the same colours. Open symbols denote positive signature and closed symbols represent negative signature.

Figure 10

Alignment and Routhian plots of the bands in ${}^{187}\mathrm{Os}$ in comparison with those in the ${}^{183}\mathrm{Os}$ and ${}^{185}\mathrm{Os}$ isotopes. Harris parameters used here are: ${\mathit{J}}_{\mathit{0}}=\text{24}{\hslash }^{\text{2}}{\mathrm{MeV}}^{-\text{1}}$ and ${\mathit{J}}_{\mathit{1}}=\text{66}{\hslash }^{\text{4}}{\mathrm{MeV}}^{-\text{3}}$ [44]. The yrast positive-parity bands are labeled by the band spin, while the excited bands are labeled as a coupling with the $\gamma$ band. The signature partner bands are labeled in the same colours, open symbols denote the positive signature and closed symbols represent the negative signature sequences.

Figure 11

Cranked shell model positive-parity Routhians for ${}^{186}\mathrm{Os}$ with N = 110 as a function of the rotational frequency, $\hslash \omega$. In this figure, solid lines represent quasiparticle trajectories with signature $\alpha =+1/2$ and dashed lines are used for quasiparticle trajectories with signature $\alpha =-1/2$.

Figure 12

Calculated excitation energies for the two lowest-energy positive-parity bands of ${}^{187}\mathrm{Os}$ in comparison with experimental data.

Figure 13

Sketch illustrating the nature of the collective excitation in QTR model. Both bands illustrated in the middle panel are based on the same single-particle configuration with projection of the angular momentum along the long axis of ${\mathrm{\Omega }}_{\ell }=\mathit{\text{11/2}}$. The angular momenta coupling for the yrast and excited bands are shown in the left and right panels, respectively. The angular momentum in the yrast band increases due to favorite rotation around the intermediate axis, and the projection of the total angular momentum along the long axis is ${\mathit{K}}_{\ell }={\mathrm{\Omega }}_{\ell }=\mathit{\text{11/2}}$. The angular momentum in the excited band includes one unit, $2\hslash$, of unfavoured rotation around the long axis, while the remaining rotation is around the intermediate axis, thus ${\mathit{K}}_{\ell }={\mathrm{\Omega }}_{\ell } +$ 2 = 15/2. The bands are labeled according to [${\mathrm{\Omega }}_{\ell }, K_{\ell }$].

Figure 14

Contributions of the wave functions labeled with their single-particle orbitals and projections $K_{\ell }$ on the long axis for the states with $\mathit{I}=\text{11/2}$–15/2 and 37/2–41/2 from the yrast, TiP1, band in ${}^{187}\mathrm{Os}$. The orbitals that contribute to the wave function, (26, 27, 28, and 29), are labeled on the x axis as in Table 2. The corresponding $K_{\ell }$ values are shown on the y-axis and also illustrated with different colours, see the legend.

Figure 15

The same as Fig. 14, but for TiP2 band.